Advanced book on Mathematics Olympiad

(ff) #1

184 3 Real Analysis


530.Letf, g:R^3 →Rbe twice continuously differentiable functions that are constant
along the lines that pass through the origin. Prove that on the unit ballB =
{(x,y,z)|x^2 +y^2 +z^2 ≤ 1 },
∫∫∫


B

f∇^2 gdV=

∫∫∫

B

g∇^2 fdV.

Here∇^2 =∂
2
∂x^2 +

∂^2
∂y^2 +

∂^2
∂z^2 is the Laplacian.

531.Prove Gauss’ law, which states that the total flux of the gravitational field through
a closed surface equals− 4 πGtimes the mass enclosed by the surface, whereGis
the constant of gravitation. The mathematical formulation of the law is
∫∫


S

−→

F ·−→ndS=− 4 πMG.

532.Let


−→
G(x, y)=

(

−y
x^2 + 4 y^2

,

x
x^2 + 4 y^2

, 0

)

.

Prove or disprove that there is a vector field

−→

F :R^3 →R^3 ,

−→

F (x, y, z)=(M(x, y, z), N (x, y, z), P (x, y, z)),

with the following properties:
(i)M, N, Phave continuous partial derivatives for all(x,y,z) =( 0 , 0 , 0 );
(ii) curl

−→

F =

−→

0 , for all(x,y,z)  =( 0 , 0 , 0 );
(iii)

−→

F (x, y, 0 )=

−→

G(x, y).

533.Let


−→

F :R^2 →R^2 ,

−→

F (x, y)=(F 1 (x, y), F 2 (x, y))be a vector field, and let
G:R^3 →Rbe a smooth function whose first two variables arexandy, and the
third ist, the time. Assume that for any rectangular surfaceDbounded by the
curveC,

d
dt

∫∫

D

G(x, y, t)dxdy=−


C

−→

F ·d

−→

R.

Prove that

∂G
∂t

+

∂F 2

∂x

+

∂F 1

∂y

= 0.
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